Interval Valued Intuitionistic Fuzzy Line Graphs

Objectives In the field of graph theory, an intuitionistic fuzzy set becomes a useful tool to handle problems related to uncertainty and impreciseness. We introduced the interval-valued intuitionistic fuzzy line graphs (IVIFLG) and explored the results related to IVIFLG. Result Some propositions and theorems related to IVIFLG are proposed and proved, which are originated from intuitionistic fuzzy graphs (IVIG). Furthermore, Isomorphism between two IVIFLGs toward their IVIFGs was determined and verified.


Introduction
After Euler was presented with the impression of Königsberg bridge problem, Graph Theory has become recognized in different academic fields like engineering, social science in medical science, and natural science. A few operations of graphs like line graph, wiener index of graph, cluster and corona operations of graph, total graph, semi-total line and edge join of graphs have been valuable in graph theory and chemical graph theory to consider the properties of boiling point, heat of evaporation, surface tension, vapor pressure, total electron energy of polymers , partition coefficients, ultrasonic sound velocity and internal energy [1][2][3][4]. The degree sequence of a graph and algebraic structure of different graphs operations were determined and its result is to the join and corona products of any number of graphs [5]. These operations are not only in classical graphs, they are more useful in fuzzy and generalizations of fuzzy graphs. The real-world problems are often full of uncertainty and impreciseness, Zadeh introduced fuzzy sets and membership degree [6]. Based on Zedeh's work, Kaufman introduced the notion of fuzzy relations [7]. Then, Rosenfeld [8] followed the Kaufman work and he introduced fuzzy graphs.
Later, Atanassov witnessed that many problems with uncertainty and imprecision were not handled by fuzzy sets(FS) [9]. Then considering this, he added the falsehood degree to membership degree and presented intuitionistic fuzzy sets(IFS) with relations and IFG which is a generalization of FS and their applications [9][10][11]. In 1993, Mordeson examined the idea of fuzzy line graphs(FLG) for the first time by proofing both sufficient and necessary conditions for FLG to be bijective homomorphism to its FG. And also some theorems and propositions are developed [12].
definitions of IVIFLG. The novelty of our works are given as follows: (1) IVIFLG is presented and depicted with an example, (2) many propositions and Theorems on properties of IVIFLG is developed and proved, (3) further, interval-valued intuitionistic weak vertex homomorphism and interval-valued intuitionistic weak line isomorphism are proposed. For the notations not declared in this manuscript, to understand well we recommend the readers to refer [10,12,14,18,19].
Definition 1 [17] The graph of the form G = (V , E) is an intuitionistic fuzzy graph (IFG) such that (i) σ 1 , γ 1 : V → [0, 1] are membership and nonmembership value of vertex set of G respectively and Definition 2 [20] The line graph L(G) of graph G is defined as i. Every vertex in L(G) corresponds to an edge in G, ii. Pair of nodes in L(G) are adjacent iff their correspondence edges e i , e j ∈ G have a common vertex v ∈ G.
is a graph with |V | = n and S i = {v i , e i 1 , · · · , e i p } where 1 ≤ i ≤ n, 1 ≤ j ≤ p i and e ij ∈ E has v i as a vertex. Then (S, T) is called intersection graph where S = {S i } is the vertex set of (S, T) and Remark The given simple graph G and its intersection graph (S, T) are isomorphic to each other(G ∼ = (S, T )). Definition 5 [16] Let I = (A 1 , B 1 ) is an IFG with A 1 = (σ A 1 , γ A 1 ) and B 1 = (σ B 1 , γ B 1 ) be IFS on V and E respectively. Then (S, T ) = (A 2 , B 2 ) is an intuitionistic fuzzy intersection graph of I whose membership and nonmembership functions are defined as ) on S and T respectively. So, IFG of the intersection graph (S, T) is isomorphic to I( means, (S, T ) ∼ = I ).
be IFS on X and E receptively. Then we define the intuitionistic fuzzy line graph L(I) = (A 2 , B 2 ) of I as are IFS on H and J respectively. The L(I) = (A 2 , B 2 ) of IFG I is always IFG. Definition 7 [16] Let I 1 = (A 1 , B 1 ) and I 2 = (A 2 , B 2 ) be two IFGs. The homomorphism of ψ :

Definition 8 [13] The interval valued FS A is characterized by
Here, σ − A (v i ) and σ + A (v i ) are lower and upper interval of fuzzy subsets A of X respectively, such that For simplicity, we used IVFS for interval valued fuzzy set.

Definition 9 Let
: v ∈ X} be IVFS. Then, the graph G * = (V , E ) is called IVFG if the following conditions are satisfied; is IVFS on V and E respectively.

Definition 10
Let G = (A 1 , B 1 ) be simple IVFG. Then we define IVF intersection graph (S, T ) = (A 2 , B 2 ) as follows: 1 A 2 and B 2 are IFS of S and T respectively, Remark The given IVFG G and its intersection graph (S,T) are always isomorphic to each other.
Definition 11 [14] An interval valued fuzzy line is defined as follows: denote a membership degree and non membership degree of ver- such that Now we start the main results of this work by introducing Interval-valued Intuitionistic Fuzzy Line Graph (IVIFLG) and providing examples.
are IVFS of H and J respectively. Then we have Then L(I) of IVIFG I is shown in Fig. 2.

the IVIFLG of I is disconnected and contradicts. Therefore, I is connected.
Conversely, suppose that I is connected IVIFG. Then, there is a path among every pair of nodes. Thus by IVI-FLG definition, edges which are adjacent in I are adjacent nodes in IVIFLG. Therefore, each pair of nodes in IVI-FLG of I are connected by a path. Hence the proof.

Proposition 19
The IVIFLG of IVIFG K 1,n is K n which is complete IVIFG with n nodes.

Proof
For IVIFG K 1,n let us take v ∈ V (K 1,n ) which is adjacent to every u i ∈ V (K 1,n ) where i = 1, 2, · · · , n . Implies that v is adjacent with every u i . Thus, in IVIFLG of K 1,n , all the vertices are adjacent. This implies that it is complete. Hence the proof.

Example 20
Consider the IVIFG K 1,3 whith vertex sets of    .

Proof
Suppose both conditions (i) and (ii) are satisfied. i.e., We know that IVIFS yields the properties will suffice. From definition of IVIFLG the converse of this statement is well known.

Proof
It is straightforward from the definition, therefore it is omitted.
Proposition 24 Let I 1 and I 2 IVIFGs of I * 1 and I * 2 respectively. If the mapping ψ : I 1 → I 2 is a weak isomorphism, then ψ : I * 1 → I * 2 is isomorphism map.

Proof
Suppose ψ : I 1 → I 2 is a weak isomorphism. Then Hence the proof.
Theorem 25 Let I * = (V , E) is connected graph and consider that L(I) = (A 2 , B 2 ) is IVIFLG corresponding to IVIFG I = (A 1 , B 1 ) . Then 1. There is a map ψ : I → L(I) which is a weak isomorphism iff I * a cyclic graph such that

2.
The map ψ is isomorphism if ψ : I → L(I) is a weak isomorphism.

Consider ψ : I → L(I) is a weak isomorphism. Then we have
This follows that I * = (V , E) is a cyclic from proposition 24. Now let v 1 v 2 v 3 · · · v n v 1 be a cycle of I * where vertices set V = {v 1 , v 2 , · · · , v n } and edges set E = {v 1 v 2 , v 2 v 3 , · · · , v n v 1 } . Then we have IVIFS and where i = 1, 2, · · · , n and v n+1 = v 1 . Thus, for t −

Theorem 26
The IVIFLG of connected simple IVIFG I is a path graph iff I is path graph. Conversely, suppose L(I) is a path graph. This implies that each degree of vertex v i ∈ I is not greater than two. If one of the degrees of vertex v i inI is greater than two, then the edges e which incident to v i ∈ I would form a complete subgraph of IVIFLG L(I) of more than two vertices. Therefore, the IVIFG I must be either cyclic or path graph. But, it can't be the cyclic graph since a line graph of the cyclic graph is the cyclic graph. Hence the proof.

Limitations
• This paper introduces only the new concept of IVI-FLG which is the extension of IFLG.
• fast, convenient online submission • We focused only on some properties of IVFLG and not all properties are mentioned. • Due to uncertainty and imprecise many real-world problems like networks communication, machine learning, data organization, traffic light control, computational devices, medical diagnosis, decision making, and the flow of computation is difficult to solve without using IFS, IVIF models it has become rapidly useful in the world. But, in this paper the application part is not included.